Identify integral: Identify the integral to be solved. We have the integral int tsqrt(1+t^(2))dt, which we need to solve.Use substitution: Use a substitution to simplify the integral. Let u = 1 + t^2. Then, du/dt = 2t, which implies that (1/2)du = tdt.Rewrite in terms of u: Rewrite the integral in terms of u. Substituting the values from the substitution, we get (1/2)int sqrt(u)du.Integrate with respect to u: Integrate with respect to u. The integral of sqrt(u) with respect to u is (2/3)u^(3/2). Therefore, (1/2)int sqrt(u)du = (1/2)*(2/3)u^(3/2) + C.Substitute back in t: Substitute back in terms of t. Since u = 1 + t^2, we have (1/2)*(2/3)(1 + t^2)^(3/2) + C = (1/3)(1 + t^2)^(3/2) + C.Write final answer: Write the final answer. The indefinite integral of tsqrt(1+t^(2)) with respect to t is (1/3)(1 + t^2)^(3/2) + C.

Calculus

Find indefinite integrals using the substitution

Full solution

\int t\sqrt{1+t^{2}}\,dt

Identify integral: Identify the integral to be solved. $\newline$ We have the integral $\int t\sqrt{1+t^{2}}\,dt$ , which we need to solve.
Use substitution: Use a substitution to simplify the integral. $\newline$ Let $u = 1 + t^2$ . Then, $\frac{du}{dt} = 2t$ , which implies that $(\frac{1}{2})du = tdt$ .
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ . Substituting the values from the substitution, we get $(1/2)\int \sqrt{u}\,du$ .
Integrate with respect to $u$ : Integrate with respect to $u$ . The integral of $\sqrt{u}$ with respect to $u$ is $\frac{2}{3}u^{\frac{3}{2}}$ . Therefore, $\frac{1}{2}\int \sqrt{u}\,du = \frac{1}{2}\cdot\frac{2}{3}u^{\frac{3}{2}} + C$ .
Substitute back in $t$ : Substitute back in terms of $t$ . $\newline$ Since $u = 1 + t^2$ , we have $(1/2)\cdot(2/3)\cdot(1 + t^2)^{3/2} + C = (1/3)\cdot(1 + t^2)^{3/2} + C$ .
Write final answer: Write the final answer. $\newline$ The indefinite integral of $t\sqrt{1+t^{2}}$ with respect to $t$ is $(\frac{1}{3})(1 + t^2)^{(\frac{3}{2})} + C$ .